@MASTERSTHESIS\{IMM2015-06922,
    author       = "K. H. Nielsen",
    title        = "The Frame Set of Gabor Systems with {B-}spline Generators",
    year         = "2015",
    school       = "Technical University of Denmark, Department of Applied Mathematics and Computer Science",
    address      = "Richard Petersens Plads, Building 324, {DK-}2800 Kgs. Lyngby, Denmark, compute@compute.dtu.dk",
    type         = "",
    note         = "{DTU} supervisor: Jakob Lemvig, jakle@dtu.dk, {DTU} Compute",
    url          = "http://www.compute.dtu.dk/English.aspx",
    abstract     = "This thesis is concerned with computational and theoretical aspects of Gabor systems in time-frequency analysis and, in particular, the representation of signals in terms of time-frequency shifts of {B-}splines constituting a frame. Frames are systems of {''}simple{''} functions or building blocks which deliver ways of analysing and representing signals in a stable manner, even in the presence of noise. Because of these desirable properties, frames play an important role in both harmonic analysis and signal processing.

One of the fundamental problems in Gabor analysis is to determine for which sampling and modulation rates, controlled by two parameters a > 0 and b > {0,} respectively, the corresponding time-frequency shifts of a given generator constitutes a frame. The so-called frame set of a generator is the parameter values (a; b) xxxxx for which the associated Gabor system generated by the generator function is a frame.

Except for the first {B-}spline very little is known about the frame set for {B-}splines. This thesis adds a considerable amount of new information on the frame set for {B-}splines. We first review some of the known characteristics of the frame set for {B-}splines. We then prove a new domain of parameter values (a; b) for which the Gabor system generated by {B-}splines is indeed a frame. Furthermore, we examine some of the unknown areas numerically, both in Matlab and Maple. From these simulations, we discover new parameter values (a; b) which do not belong the frame set of the {B-}splines of order two. This, in turn, disproves a recent conjecture by Karlheinz Grochenig. Finally, we formulate two new conjectures on the frame set of the {B-}splines of order two based on our numerical and theoretical findings."
}